#6467092

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2036 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,21)(17,31)(18,24)(19,23)(22,29)(25,35)(26,39)(27,38)(28,37)(30,40)(32,44)(33,41)(34,42)(36,43);(2,4)(13,15)(14,20)(16,26)(17,32)(18,34)(19,28)(21,39)(22,35)(23,37)(24,42)(25,29)(27,33)(30,43)(31,44)(36,40)(38,41);(6,9)(10,12)(13,16,18,29,19,30,33,17)(14,21,24,22,23,40,41,31)(15,25,27,26,28,32,34,43)(20,35,38,39,37,44,42,36);(13,17,33,30,19,29,18,16)(14,22,41,21,23,31,24,40)(15,26,34,25,28,43,27,32)(20,36,42,44,37,39,38,35);(5,7,8,11)(6,10,9,12)(14,23)(20,37)(21,40)(22,31)(24,41)(35,44)(36,39)(38,42)

Polycyclic presentation:

2^9|c^2=h,d^2=hi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=dh,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374288);